Week2bEECS 42, Spring 2005 Prof. White

Find i2, i1 and io

Circuit w/ Dependent Source Example

Week2bEECS 42, Spring 2005 Prof. White

Simplify a circuit before applying KCL and/or KVL:

+7 V

Using Equivalent Resistances

R1 = R2 = 3 kR3 = 6 k

R4 = R5 = 5 kR6 = 10 k

I

R1

R2

R4

R5

R3

R6

Example: Find I

Week2bEECS 42, Spring 2005 Prof. White

The Wheatstone Bridge

• Circuit used to precisely measure resistances in the range from 1 to 1 M, with ±0.1% accuracy R1 and R2 are resistors with known values

R3 is a variable resistor (typically 1 to 11,000)

Rx is the resistor whose value is to be measured

+V–

R1 R2

R3 Rx

current detector

battery

variable resistor

Week2bEECS 42, Spring 2005 Prof. White

Finding the value of Rx

• Adjust R3 until there is no current in the detector

Then,

+V–

R1 R2

R3 Rx

Rx = R3

R2

R1Derivation:

i1 = i3 and i2 = ix

i3R3 = ixRx and i1R1 = i2R2

i1R3 = i2Rx

KCL =>

KVL =>

R3

R1

Rx

R2

=

i1 i2

ixi3

Typically, R2 / R1 can be varied from 0.001 to 1000 in decimal steps

Week2bEECS 42, Spring 2005 Prof. White

Some circuits must be analyzed (not amenable to simple inspection)

-

+ R2

R1

V

I

R4

R3

R5

Special cases:R3 = 0 OR R3 =

R1

+

R4R5

R2

V R3

Identifying Series and Parallel Combinations

Week2bEECS 42, Spring 2005 Prof. White

Resistive Circuits: Summary

• Equivalent resistance of k resistors in series:

Req = Ri = R1 + R2 + ••• + Rk

• Equivalent resistance of k resistors in parallel:

• Voltage divided between 2 series resistors:

• Current divided between 2 parallel resistors:

R2

R1

vs

I

+

+v1

–

R2

R1

vs

I

++

+v1

–

i=1

k

1

Req

= ••• + i=1

k 1

Ri

1

R1

1

R2

1

Rk

v1 = vs

R1

R1 + R2

i1 = isR2

R1 + R2

R2R1iS

i2i1

R2R1iS

i2i1

Week2bEECS 42, Spring 2005 Prof. White

Circuit Analysis Methods …

Given: a circuit with voltage and/of current sources, and components

(Rs, Cs, Ls, diodes, transistors, etc.) connected by wires

Task: find all the voltages and currents of interest

Approaches:

1. Use Ohm’s Law, KCL and KVL piecemeal

2. Nodal analysis – node voltages are the unknowns

3. Mesh (loop) analysis – mesh currents are the unknowns

4. Superposition (linear circuits) – work with only one voltage or current source at a time) and use #1, #2 or #3

5. Computer simulation of circuit behavior

Week2bEECS 42, Spring 2005 Prof. White

1. Choose a reference node (“ground”)Look for the one with the most connections!

2. Define unknown node voltagesthose which are not fixed by voltage sources

3. Write KCL at each unknown node, expressing current in terms of the node voltages (using the I-V relationships of branch elements)

Special cases: floating voltage sources

4. Solve the set of independent equations N equations for N unknown node voltages

Node-Voltage Circuit Analysis Method

Week2bEECS 42, Spring 2005 Prof. White

1. Choose a reference node.

2. Define the node voltages (except reference node and the one set by the voltage source).

3. Apply KCL at the nodes with unknown voltage.

4. Solve for unknown node voltages.

R4 V1 R2

+ - IS

R3R1

Nodal Analysis: Example #1

Week2bEECS 42, Spring 2005 Prof. White

Week2bEECS 42, Spring 2005 Prof. White

V2V1

R2

R1

R4

R5

R3 I1

Va

Nodal Analysis: Example #2

Week2bEECS 42, Spring 2005 Prof. White

Week2bEECS 42, Spring 2005 Prof. White

A “floating” voltage source is one for which neither side is connected to the reference node, e.g. VLL in the circuit below:

Problem: We cannot write KCL at nodes a or b because there is no way to express the current through the voltage source in terms of Va-Vb.

Solution: Define a “supernode” – that chunk of the circuit containing nodes a and b. Express KCL for this supernode. Incorporate voltage source constraint into KCL equation.

R4R2 I2

Va Vb

+ -

VLL

I1

Nodal Analysis w/ “Floating Voltage Source”

Week2bEECS 42, Spring 2005 Prof. White

supernode

Eq’n 1: KCL at supernode

Substitute property of voltage source:

R4R2 I2

Va Vb

+ -

VLL

I1

Nodal Analysis: Example #3

Week2bEECS 42, Spring 2005 Prof. White

Week2bEECS 42, Spring 2005 Prof. White

Node-Voltage Method and Dependent Sources

• If a circuit contains dependent sources, what to do?

Example:

–+

–+

80 V5i

20 10

200 2.4 A

i

Week2bEECS 42, Spring 2005 Prof. White

Node-Voltage Method and Dependent Sources

• Dependent current source: treat as independent current source in organizing and writing node eqns, but include (substitute) constraining dependency in terms of defined node voltages.

• Dependent voltage source: treat as independent voltage source in organizing and writing node eqns, but include (substitute) constraining dependency in terms of defined node voltages.

Week2bEECS 42, Spring 2005 Prof. White

–+

–+

80 V5i

20 10

200 2.4 A

i

Example: